Entropy Generation through the Interaction of Laminar Boundary-Layer Flows: Sensitivity to Initial Conditions

A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to the interaction of a longitudinal vortex with a laminar boundary-layer flow. These three-dimensional equations are cast into a Lorenz-format system of equations for the spectral velocity component solutions. Tsallis-form empirical entropic indices are obtained from the solutions of the modified Lorenz equations. These solutions are sensitive to the initial conditions applied to the time-dependent coupled, non-linear differential equations for the spectral velocity components. Eighteen sets of initial conditions for these solutions are examined. The empirical entropic indices yield corresponding intermittency exponents which then yield the entropy generation rates for each set of initial conditions. The flow environment consists of the flow of hydrogen gas with impurities at a given temperature and pressure in the interaction of a longitudinal vortex with a laminar boundary layer flow. Results are presented that indicate a strong correlation of predicted entropy generation rates and the corresponding applied initial conditions. These initial conditions may be ascribed to the turbulence levels in the boundary layer, thus indicating a source for the subsequent entropy generation rates by the interactive instabilities.


Introduction
Results are reported for an innovative computational procedure applied to a study of the sensitivity to initial conditions in the computation of entropy gen-How to cite this paper: Isaacson, L.K. (2018) Entropy Generation through the The nonlinear, time-series solutions for the spectral velocity wave components are obtained from a modified Lorenz-type set of equations that is sensitive to the initial conditions applied to the integration of the equations. The control parameters for these equations are the steady state boundary layer velocity gradients that are determined by the particular value of the kinematic viscosity for the system. Experimental measurements of the unsteady fluctuation levels in laminar boundary layers when subjected to free stream turbulence have been presented by Walsh and Hernon [6]. These results indicate that the free stream turbulence level has a significant effect on the resulting entropy generation rates in laminar boundary layers. The initial conditions for the integration of the Lorenz-type equations are heuristically assumed to be attenuated levels of the turbulence imposed on the system from the free stream.
However, the initial conditions for the integration of the modified Lorenz equations are the actual turbulent intensity levels applied to the flow system at the specific location within the boundary layer where the computational results are obtained. We have selected eighteen sets of initial conditions for the initial values for the computation of the time development of the spectral stream wise, normal and span wise velocity components.
The boundary-layer environment used in the study reported here is the flow of helium with slight impurities at a temperature of T = 320.0 K and a pressure of p = 1.01325 × 10 5 N/m 2 at a normalized boundary-layer vertical location of η = 3.00. The kinematic viscosity for the helium mixture at these conditions is ν = 1.384696 × 10 −4 m 2 /s. The Falkner-Skan transformation, in the form 1 2 e u y x η ν   =     (1) provides the definition of the normalized distance, η from the surface of the boundary layer flow. In this expression, u e is the boundary layer edge velocity, x is the stream wise distance and the edge value for the normalized vertical distance is η ∞ = 8.0.
This article includes the following sections: In Section 2, the laminar boundary-layer flow configuration considered in this study is described. In Section 3, eighteen sets of initial conditions for the integra- Singer [8] has reported the results of the direct numerical simulation of the effect of strong free stream turbulence on the development of a young turbulent spot in laminar boundary layer flow. These studies indicate the development of a counter-clockwise stream wise vortex that produces a laminar boundary layer in the z-y plane of the flow environment as shown in Figure 1. Ersoy and Walker [9] discus the development of this z-y plane boundary layer produced by the interaction of the vortex tangential velocity with the flow surface. Belotserkovskii and Khlopkov [10] have computed the normalized span wise velocity at the outer edge of the vortex structure as w e = 0.08. This is the value we use for the span wise velocity. Schmid  The computer source code listings that we have used to compute the steady laminar boundary layer velocity profiles for both the x-y plane and the z-y plane were developed by Cebeci and Bradshaw [12] and Cebeci and Cousteix [13].
These orthogonal profiles are similar in nature (Hansen [14]) and thus form the steady boundary layer velocity gradient control parameters for the solution of the modified Lorenz equations. The working gas for these studies is a mixture of helium with several impurities, as described in [3], at a temperature of 320.0 K and a pressure of 0.101325 MPa.
The boundary layer profiles are determined at six stream-wise stations, with the first station designated as the transmitter station. The following five stations are designated as receiver stations, with the first receiver station designated as station 1. The results obtained for the entropy generation rates at the receiver station 3, at x = 0.120, as a function of the initial conditions, are presented in depth in this article. A computational flow chart is presented in Figure 2 for the overall path of the computational procedure. The primary result of this study is the strong correlation of the resultant entropy generation rates with the corresponding applied initial conditions for those rates. This is discussed in the next section.

Selection of Initial Conditions
An essential aspect of the computational procedure discussed in this article is the    Table   1 shows the values for eighteen sets of initial conditions over this range. Also included in Table 1 are corresponding values of an equivalent turbulence level as defined by Sengupta (pp. 103-105 [15]). Each of these set numbers are shown in Figure 3, for the corresponding entropy generation rates.

Entropy Generation Rates: Sensitivity to Initial Conditions
Application of the computational procedure outlined in Figure 2 to the fluctuating velocity components in the three-dimensional flow shown in Figure 1 yields the entropy generation rates that occur through the dissipation of the or-   generation rate for each set of initial conditions shown in Table 1 has been computed for a three-dimensional boundary-layer flow of a helium mixture at a temperature of T = 320.0 K and a pressure of p = 0.101325 MPa. This temperature and pressure for this mixture provide a kinematic viscosity of v = 1.384696 × 10 −4 m 2 /s. Figure 3 shows the entropy generation rate at the stream wise station x = 0.120, for a normalized boundary layer distance of η = 3.00 [1] for each of the sets of initial conditions listed in Table 1.
These results indicate a strong correlation of the predicted entropy generation rates with the corresponding initial conditions applied to the equations for the time dependent solutions. If the assumption is made that these initial conditions are provided by the attenuated free stream turbulence, these computational methods may then provide a path to understanding bypass transition. A detailed explanation of the procedures used in the calculation of these results and a discussion of the possible sources for the indicated entropy generation rates are presented in the next sections.

Transformation of the Townsend Equations to the Modified Lorenz Format
For the flow of a wall shear layer with velocity fluctuations, the computational procedure may be separated into the evaluation of the steady state velocity profiles and a set of equations for the fluctuating velocity field (Townsend [1]). The non-equilibrium spectral equations of Townsend [1] and Hellberg, et al. [7] are arranged into a Lorenz format (Sparrow [2]) for the computation of the nonlinear time series solutions for the fluctuating spectral velocity field. The time-dependent spectral equations of Townsend [1] and Hellberg, et al. [7] are then solved with the steady state boundary layer velocity profiles as control parameters. The solutions of the modified Lorenz equations yield the spectral velocity components within the nonlinear time series solutions. Statistical analysis of these spectral time-series solutions yields the power spectral densities and the empirical entropies over a range of sixteen empirical modes. The correspondence of the peaks of the spectral power spectrum and the empirical modes of the singular value decomposition analysis is achieved by invoking the Weiner-Khintchine theorem (Attard (pp. 354-355 [16]). This theorem relates the power density spectrum and the autocorrelation function since both properties are computed from the same nonlinear time series data.
The equations for the velocity fluctuations within a wall shear layer may be written as (Townsend (pp. 46-49 [1])): Fourier transform of these equations yields the equations for the three The nonlinear coupling terms in the spectral velocity components in Equations (3) are represented in our series of equations by characterizing the transfer is introduced to provide the proper weighting of the transfer matrix (Equation (4) in our computational procedure. K is an empirical amplitude factor [18] and ( ) k t is given by: Substitution of Equation (5) The expressions for the coefficients yn σ , xn σ , r n , s n , and b n are given in detail in [3]. These coefficients are functions of the wave number components, k i , and

Synchronization Properties of the Modified Lorenz Equations
We apply the transformation of the pattern matrix (Equation (5) These instabilities are then transferred to the next station, or first receiver station. The modified Lorenz equations have been shown to have the property of synchronization or extraction of ordered signals from a chaotic signal. We will apply this property to each of the receiver stations in the system.
The synchronization properties of systems of Lorenz-type equations have been shown by Pecora and Carroll [19], Pérez and Cerdeira [20], and Cuomo and Oppenheim [21] to have the capability to extract messages masked by chao- For each receiver station, n, the system of nonlinear dynamic equations is written as: Note that for the initial station, characterized as the transmitter station, a x0 , is the time-dependent spectral velocity wave component output from the transmitter station. The input driving term for the next station, the first receiver station at x = 0.08, is then given by a x0 , where a x0 is the output from the initial or transmitter station at x = 0.06. In Equations (10-12), the input driving signal, a rn, carrying information from the transmitter and the previous receiver stations to the n-th station is given by the sum of the outputs from the transmitter station and the previous n-1 receiver stations: The initial conditions for the fluctuating spectral velocity wave vector components for the transmitter station and for each successive receiver station are set equal to the values listed in Table 1. This process determines the result that the outputs from each of the receiver stations will be masked by the original transmitter output signal, and that the synchronization process will yield an indication of the ordered regions within the transmitter signal and the output signal from each of the receiver stations.

Sensitivity to Initial Conditions
The free stream velocity for the stream wise boundary layer flow is taken as unity, u e = 1.00, while the vortex tangential velocity is w e = 0.08 (pp. 101-102 [10]).  Table 1.
The solutions of nonlinear, coupled differential equations, such as the modified Lorenz equations (Equations (7-10) and Equations (10-12)), are sensitive to the particular values of the initial conditions applied in the solutions. Table 1 presents a range of initial conditions for the solutions of these couple equations

Deterministic Results for the Modified Lorenz Equations
The solutions of the modified Lorenz equations have been obtained for each set of intial conditions listed in Table 1 for the designated stream wise stations. We have chosen to present graphical results in Figure 4 for initial conditions Set 8, Figure 5 for Set 9 and Figure 6 for Set 10. These results are obtained for a flow temperature of T = 320.0 K and a pressure of p = 0.101325 MPa. Figure 4 shows the phase diagram for a y3 − a x3 , where a y3 is the normal spectral velocity wave    Figure 6 shows the phase diagram for a y3 − a z3 , where a z3 is the span wise spectral velocity wave component and a y3 is the normal spectral velocity wave component, again at the station x = 0.120. These results indicate the formation of an initially strong spiral cone in the stream wise direction, transforming into a strongly oscillating motion in the stream wise, normal and span wise spectral planes of the flow environment.
In Figure 3

Power Spectral Density Empirical Modes
Burg's method (Chen [22]) is used to compute the power spectral densities The first five sets of initial conditions in Table 1 Table 1, strong instabilities are observed in both the normal and the span wise components of the spectral velocity components. Figure 7 presents the power spectral density for the normal spectral velocity wave component, a y3 at the third receiver station at x = 0.120, for initial conditions, Set 6. For the power spectral density spectrum, we have assigned empirical mode numbers to the peaks, starting with mode j = 1 representing the highest peak in the distribution, continuing to mode j = 16, representing the corresponding lowest peak among the sixteen peaks.
The power spectral density for the normal spectral velocity component shown in Figure 7 indicates that the kinetic energy available for dissipation is distributed in well-defined spectral peaks or empirical modes. The kinetic energy within each empirical mode, j ξ , of the power spectral density distribution is computed using Simpson's integration rule. The sum of the individual contributions across the modes then yields the total kinetic energy contained within the ordered regions. This value is then used to get the fraction of kinetic energy in each mode that is available for dissipation into internal energy.  [23] is made up of two parts, the computation of the autocorrelation matrix and the singular value decomposition of that matrix. This computer program then yields the empirical eigenvalues for each of the empirical eigenfunctions for the given nonlinear time series data segment. The singular value decomposition of the nonlinear time series solutions of the modified Lorenz equations yields the distribution of the spectral velocity component eigenvalues λ j across the empirical modes, j, for each set of initial conditions listed in Table 1. Using Parseval's theorem, (Thomas (pp. 97-100 [26])), the eigenvalues in the spectral plane, λ j , are equivalent to the eigenvalues in the physical plane. These eigenvalues therefore represent the distribution of the kinetic energy of the fluctuating velocity components across the empirical modes, j. The fractional eigenvalues have an approximate exponential distribution over the empirical modes, j, as shown in Figure 8 (Isaacson [27]).

Singular Value Decomposition and Empirical Entropies
Therefore, the analysis of Rissanen (pp. 58-60 [28]) is applicable and the empirical entropy, Semp j , may be obtained from these eigenvalues by the expression: In this expression, λ j is the empirical eigenvalue computed from the singular value decomposition procedure applied to the nonlinear time-series solution. Journal of Modern Physics The results for the empirical entropy value for each of the empirical modes allows us to compute a corresponding entropic index for these modes. This process is described in the next section.

Empirical Entropic Indices
The power spectral density spectrum shown in Figure 7 indicates regions of strongly peaked kinetic energy densities. The application of the singular value decomposition of the given time series data also provides us with a corresponding value of the empirical entropy for each peak. These two properties allow us to construct a computational method to follow these regions from ordered structures into equilibrium thermodynamic states. To accomplish this, we use the approach of the Tsallis entropic indices (Tsallis [29]). Journal of Modern Physics The empirical entropy, Semp j describes the entropy of an ordered region identified by the empirical eigenvalue, j λ , for the empirical mode, j. We have found that an expression from which we may extract an empirical entropic index, q j , from the empirical entropy, Semp j , may be written in a Tsallis entropic format as [18]: The empirical entropy index, q j , provides a connection between the empirical entropy obtained from the singular value decomposition to the intermittency exponent of the ordered structures within the time series solutions. The intermittency exponent describes the fraction of the available kinetic energy within each empirical mode, j, that is dissipated into thermodynamic internal energy, thus increasing the entropy of the system. The intermittency exponent for each of the empirical modes is discussed in the next section.

Empirical Intermittency Exponents
The final phase of the dissipation of fluctuating kinetic energy into thermodynamic internal energy occurs through the process of intermittency exponents and a relaxation process into the final thermodynamic entropy state.
The intermittency exponents for the each of the empirical modes, ζ j , are obtained from the empirical entropic indices of the Tsallis form extracted from the empirical entropies in Equation (15). Arimitsu and Arimitsu [30] derived, using multifractral methods, a relationship from which the intermittency exponent, ζ j , may be extracted from the entropic index of Tsallis. The intermittency exponent provides the fraction of fluctuating kinetic energy within the non-equilibrium empirical mode, j, that is dissipated into thermodynamic internal energy [30]. The absolute value of the empirical entropic index calculated from Equation (15) is used to extract the intermittency exponent from the equation derived by Arimitsu and Arimitsu [30]. This expression for the empirical mode, j, is written as: The intermittency exponent, ζ j , found from this expression, represents the fraction of kinetic energy in the empirical mode, j, that is dissipated into background thermal energy. The kinetic energy contained within the spectral mode, j, of the power spectral density is denoted as ξ j . Thus, the product of the kinetic energy of the mode, j, and the intermittency exponent for that mode, ζ j , summed over all of the empirical modes, represents the amount of kinetic energy in the given spectral velocity component that is dissipated into increasing the entropy of the reservoir. Journal of Modern Physics The computation of the intermittency exponents yields two significant results.
First, the computed value for each empirical mode allows the computation of the entropy generated through the dissipation of that mode. Second, the particular value for each empirical mode provides us with additional insight into the physical processes involved in the generation of entropy through the dissipation of the empirical modes embedded in the nonlinear solutions of the modified Lorenz equations.
Consider the entropy generation rates for initial conditions Sets 8, 9 and 10 in Figure 3. The entropy generation rate for Set 9 indicates a spike in the value of the generation rate compared to the rates generated for Sets 8 and 10. We wish to compare the intermittency exponents for each of these sets of initial conditions to better understand the physical processes related to the generation of the spike for Set 9. Note that the stream wise intermittency is relatively low for all three sets of initial conditions. The increase in value for the stream wise intermittency exponents at the high empirical modes do not make a signifiacant contribution Figure 9. The intermittency exponents are shown for the spectral velocity components, a x3 , a y3 , and a z3 , for the initial conditions of Set 8 at the stream wise location of x = 0.120, the normalized vertical location of η = 3.00 and the span wise velocity of w e = 0.080. Figure 10. The intermittency exponents are shown for the spectral velocity components, a x3 , a y3 , and a z3 , for the initial conditions of Set 9 at the stream wise location of x = 0.120, the normalized vertical location of η = 3.00 and the span wise velocity of w e = 0.080. Figure 11. The intermittency exponents are shown for the spectral velocity components, a x3 , a y3 , and a z3 , for the initial conditions of Set 10 at the stream wise location of x = 0.120, the normalized vertical location of η = 3.00 and the span wise velocity of w e = 0.080. Journal of Modern Physics to the entropy generation rates because these modes contain very low fractions of the available stream wise kinetic energy.
However, for the normal and span wise spectral velocity components for initial conditions Sets 8 and 10, the first and third empirical modes make signifant contributions to the values of the intermittency exponents, with the third mode dominating the contributions. For the initial conditions Set 9, the fifth empirical mode comes into play with a significant increase in value. This indicates that as the initial conditions are increased in magnitude, the nonlinear time dependent solution of the modified Lorenz equations for the normal spectral velocity component predicts the spread of ordered kinetic energy over an additional empirical mode for this component. This results in the predition of a higher rate of entropy generation for this particular set of initial conditions. The repeated pattern shown in Figure 3 for the generation rates over the first twelve sets of initial conditions is rather surprising, given that we are using coupled, nonlinear first-order differential equations in our modified Lorenz equations. Figure 12 shows the intermittency exponents for the three spectral velocity components for initial conditions Set 15, which is the maximum entropy generation rate shown in Figure 3. The stream wise intermittency exponent shows an increase for empirical modes 1 and 3, with the span wise component increasing for modes 1, 3, and 5. The normal component indicates an almost linear increase in intermittency exponent value over empirical modes 1, 3, 5, and 7. This is also an interesting pattern within the nonlinear solutions of the modified Lorenze equations with the increase in intial conditions.
The foundation for the Tsallis entropic index [29] and the Arimitsu and Arimitsu intermittency exponent [30] lies in the concept of the fractal nature of the dissipation of turbulent kinetic energy (Mandelbrot [31]). Mandelbrot [31] also introduced the concept of fractals to describe the geometry of turbulent intermittency. Frisch and Parisi [32] noted that there are actually many fractal scales involved in the dissipation of turbulent energy and in the process of intermittency and introduced the concept of the multifractal model of turbulence. We have taken advantage of the considerable progress that has been made in extending these models to actual physical processes and wish to compare our results with recent theoretical and experimental results concerning the intermittency of the turbulent dissipation of kinetic energy.
To compare our computational results with results presented in the literature, we need to obtain average values for the intermittency exponents computed for a selected set of initial conditions from Table 1. For example, we will start with the results obtained for the initial conditions in Set 5 of Table 1. This set of initial conditions did not indicate the generation of significant flow instabilities and has a relatively low entropy generation rate, as indicated in Figure 3. The average intermittency exponent for the set of initial conditions is found by first averaging the intermittency exponents found for each empirical mode of the singular value decomposition procedure applied to each of the three spectral velocity components. Then, these three average values are averaged across the three Journal of Modern Physics Figure 12. The intermittency exponents are shown for the spectral velocity components, a x3 , a y3 , and a z3 , for the initial conditions of Set 15 at the stream wise location of x = 0.120, the normalized vertical location of η = 3.00 and the span wise velocity of w e = 0.080. spectral velocity components to yield the average intermittency exponent for the entire Set 5 initial conditions. The resulting intermittency exponent will be designated as 5ave ζ . The value of the average intermittency exponent for initial conditions Set 5 is found to be 5ave ζ = 0.349. Arimitsu and Arimitsu [33] found, in the analysis of quantum turbulent intermittency, a value of 5ave ζ = 0.326, while Arimitsu, et al. [34] found by DNS analysis of this same quantum system a value of 5ave ζ = 0.345. We may be able to gain a better understanding of the fundamental characteristics of systems with high values of intermittency exponents through a comparison of these two different systems. The low-temperature quantum superfluid system studied in [33] [34] appears to have negligible mutual friction between the superfluid and the normal fluid components. Therefore, the level of irreversibilities produced in the system would be very low. In the three-dimensional boundary-layer with initial conditions given in Set 5 of Table 1, very low levels of instabilities are found in the nonlinear time-series solutions of the modified Lorenz equations. Subsequently, low values are predicted for the entropy generation rates for the initial conditions of Set 5. Thus, the similarity between these two systems is that they each have very low levels of irreversibilities.
However, when we move to the initial conditions Set 6, the first significant instability is found in the nonlinear time series solutions of the modified Lorenz Meneneau and Sreenivan [35]. Table 2 shows the average intermittency exponents for a selected range of initial conditions, corresponding to the patterns apparent in Figure 3.  [36]. It should also be noted that, in an analysis of the number of steps in the cascade process of the dissipation of turbulent kinetic energy [36], the number of steps was found to be 16.4, which is close to the number of empirical modes, 16, that we have found from the power spectral density results.

Kinetic Energy Available for Dissipation
The source of the kinetic energy to be dissipated through the empirical modes is considered as the local steady-flow kinetic energy, u 2 /2, at the normalized vertical distance, η = 3.0 in the x-y plane boundary layer. This kinetic energy is assumed to be distributed over the three fluctuating velocity components. The fraction of kinetic energy in the x-direction velocity component is denoted as and span wise velocity components is the summation, over the empirical modes, j, of the product of the kinetic energy fraction of each mode, j ξ , times the intermittency exponent for that mode, ϕ ξ [3].
The empirical intermittency exponent for each of the empirical modes has been obtained from the expression (Equation (16)) given by Arimitsu and Arimitsu [31]. Thus, values are available for the input energy source for the non-equilibrium ordered regions, the fraction of the fluctuation kinetic energy  [33] available in each of the empirical modes within the non-equilibrium ordered regions, and the fraction of the energy in each of the empirical modes that dissipates into background thermal energy, thus increasing the thermodynamic entropy. Concepts from non-equilibrium thermodynamics are used to calculate the dissipation process for the ordered regions as a general relaxation process. This is considered in the next section.

Entropy Generation Rates through the Dissipation of Ordered Regions
de Groot and Mazur (pp. 221-230 [37]), from the concepts of non-equilibrium thermodynamics, write the equation for the entropy generation rate in an internal relaxation process as: Here, s is the entropy per unit mass, μ is the mechanical potential for the transport of the ordered regions in an external context and J(x) is the flux of kinetic energy through the ordered regions available for dissipation into thermal internal energy. The dissipation of the ordered regions into background thermal energy may be considered as a two-stage process from the transition of the or- In this expression, ρ is the density of the working substance, T is the temperature and u e is the free stream velocity. The dissipation rate for each of the three fluctuating spectral velocity components is included in Equation (18).
The kinetic energy in each spectral mode available for final dissipation into equilibrium internal energy is computed for each of the spectral peaks shown in Figure 7. The empirical entropy for each of the regions indicated by the spectral peaks is found from the singular value decomposition process applied to the given time series data segment. The connecting parameter, the empirical entropic index, is then extracted from the resulting value of the empirical entropy.
The empirical entropic indices then allow the extraction of the corresponding intermittency exponents.

Discussion
There are two significant issues with the computational procedure and the results reported in this article. First, there have been no comparable computational results which would validate the procedures adopted here. Second, experimental validation is sparse and applies only to selected aspects of the computational approach and the results. However, the computational procedure is innovative in that it provides a method for the incorporation of a deterministic set of equations for the development of instabilities within the steady state environment of a three-dimensional laminar boundary layer flow. The results indicate that the entropy generation rates resulting from nonlinear interactions in a threedimensional laminar boundary-layer flow are significantly affected by the particular initial conditions that are applied to the longitudinal vortex structure and the adjacent laminar boundary layer flow.
The counter clockwise rotating longitudinal vortex structure creates a viscous boundary layer along the z-y plane of the flow configuration. This viscous boundary layer is orthogonal to the laminar boundary layer in the x-y plane in the stream wise direction. It is shown that this nonlinear interaction creates instabilities within the three-dimensional flow configuration.
The computational results reported here for the entropy generation rates for a helium mixture boundary layer flow are obtained at the stream wise location of x = 0.120, in the range of stream wise locations from x = 0.06 to x = 0.18, for the normalized vertical station of η = 3.00. The weighting factor K in Equation (5) has been found to yield the prediction of instabilities for a value of K = 0.05. In an experimental investigation of laminar boundary layer receptivity to surface mass injection, Sengupta (pp. 158-170 [15]) found that the coefficient of 0.05 for a time-dependent sinusoidal surface mass injection rate also initiated instabilities within a laminar boundary-layer flow. We thus have the implication that Journal of Modern Physics Equation (5) is a proper choice for the transformation of the Townsend equations for the fluctuating velocity components into a modified Lorenz format.
Free stream turbulence levels provide the turbulent kinetic energy that enters the wall shear layer. However, this level is attenuated within the layer and only a portion is available to serve as the initial conditions for the solution of the mod- The sensitivity to initial conditions of the Lorenz format spectral velocity equations may provide a means of connecting the incorporation of these time dependent spectral equations in the computational procedure with the concept of bypass transition of the boundary layer flow due to outside disturbances.

Conclusions
The These results offer a deterministic path to the understanding of bypass transition and a foundation for the development of an understanding of the dynamics of turbulent spots in the transition from laminar to turbulent flows.

Conflicts of Interest
The author declares no conflict of interest.